The geometry of dimer models
David Cimasoni1
1 Université de Genève, Section de mathématiques, 2-4 rue du Lièvre, 1211 Genève 4, Switzerland
Winter Braids Lecture Notes, Volume 1 (2014), Talk no. 2, 14 p.

This is an expanded version of a three-hour minicourse given at the winterschool Winterbraids IV held in Dijon in February 2014. The aim of these lectures was to present some aspects of the dimer model to a geometrically minded audience. We spoke neither of braids nor of knots, but tried to show how several geometric tools that we know and love (e.g. (co)homology, spin structures, real algebraic curves) can be applied to very natural problems in combinatorics and statistical physics. These lecture notes do not contain any new results, but give a (relatively original) account of the works of Kasteleyn [14], Cimasoni-Reshetikhin [4] and Kenyon-Okounkov-Sheffield [16].

Published online:
DOI: 10.5802/wbln.3
David Cimasoni 1

1 Université de Genève, Section de mathématiques, 2-4 rue du Lièvre, 1211 Genève 4, Switzerland 
@article{WBLN_2014__1__A2_0,
     author = {David Cimasoni},
     title = {The geometry of dimer models},
     journal = {Winter Braids Lecture Notes},
     note = {talk:2},
     publisher = {Winter Braids School},
     volume = {1},
     year = {2014},
     doi = {10.5802/wbln.3},
     mrnumber = {3703249},
     zbl = {1426.82005},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/wbln.3/}
}
TY  - JOUR
AU  - David Cimasoni
TI  - The geometry of dimer models
JO  - Winter Braids Lecture Notes
N1  - talk:2
PY  - 2014
VL  - 1
PB  - Winter Braids School
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/wbln.3/
DO  - 10.5802/wbln.3
LA  - en
ID  - WBLN_2014__1__A2_0
ER  - 
%0 Journal Article
%A David Cimasoni
%T The geometry of dimer models
%J Winter Braids Lecture Notes
%Z talk:2
%D 2014
%V 1
%I Winter Braids School
%U https://proceedings.centre-mersenne.org/articles/10.5802/wbln.3/
%R 10.5802/wbln.3
%G en
%F WBLN_2014__1__A2_0
David Cimasoni. The geometry of dimer models. Winter Braids Lecture Notes, Volume 1 (2014), Talk no. 2, 14 p. doi : 10.5802/wbln.3. https://proceedings.centre-mersenne.org/articles/10.5802/wbln.3/

[1] Cahit Arf. Untersuchungen über quadratische Formen in Körpern der Charakteristik 2. I. J. Reine Angew. Math., 183:148–167, 1941. | DOI | Zbl

[2] Michael F. Atiyah. Riemann surfaces and spin structures. Ann. Sci. École Norm. Sup. (4), 4:47–62, 1971. | DOI | MR | Zbl

[3] David Cimasoni. Dimers on graphs in non-orientable surfaces. Lett. Math. Phys., 87(1-2):149–179, 2009. | DOI | MR | Zbl

[4] David Cimasoni and Nicolai Reshetikhin. Dimers on surface graphs and spin structures. I. Comm. Math. Phys., 275(1):187–208, 2007. | DOI | MR | Zbl

[5] David Cimasoni and Nicolai Reshetikhin. Dimers on surface graphs and spin structures. II. Comm. Math. Phys., 281(2):445–468, 2008. | DOI | MR | Zbl

[6] N. P. Dolbilin, Yu. M. Zinov’ev, A. S. Mishchenko, M. A. Shtan’ko, and M. I. Shtogrin. Homological properties of two-dimensional coverings of lattices on surfaces. Funktsional. Anal. i Prilozhen., 30(3):19–33, 95, 1996. | DOI

[7] Jack Edmonds. Optimum branchings. J. Res. Nat. Bur. Standards Sect. B, 71B:233–240, 1967. | DOI | MR

[8] Michael E. Fisher. Statistical mechanics of dimers on a plane lattice. Phys. Rev., 124(6):1664–1672, Dec 1961. | DOI | MR | Zbl

[9] Anna Galluccio and Martin Loebl. On the theory of Pfaffian orientations. I. Perfect matchings and permanents. Electron. J. Combin., 6:Research Paper 6, 18 pp. (electronic), 1999. | DOI | MR

[10] Axel Harnack. Ueber die Vieltheiligkeit der ebenen algebraischen Curven. Math. Ann., 10(2):189–198, 1876. | DOI | MR | Zbl

[11] Dennis Johnson. Spin structures and quadratic forms on surfaces. J. London Math. Soc. (2), 22(2):365–373, 1980. | DOI | MR | Zbl

[12] P. W. Kasteleyn. The statistics of dimers on a lattice. Physica, 27:1209–1225, 1961. | DOI | Zbl

[13] P. W. Kasteleyn. Dimer statistics and phase transitions. J. Mathematical Phys., 4:287–293, 1963. | DOI | MR

[14] P. W. Kasteleyn. Graph theory and crystal physics. In Graph Theory and Theoretical Physics, pages 43–110. Academic Press, London, 1967. | Zbl

[15] Richard Kenyon and Andrei Okounkov. Planar dimers and Harnack curves. Duke Math. J., 131(3):499–524, 2006. | DOI | MR | Zbl

[16] Richard Kenyon, Andrei Okounkov, and Scott Sheffield. Dimers and amoebae. Ann. of Math. (2), 163(3):1019–1056, 2006. | DOI | MR | Zbl

[17] L. Lovász and M. D. Plummer. Matching theory, volume 121 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1986. Annals of Discrete Mathematics, 29. | DOI | Zbl

[18] G. Mikhalkin. Real algebraic curves, the moment map and amoebas. Ann. of Math. (2), 151(1):309–326, 2000. | DOI | MR | Zbl

[19] Grigory Mikhalkin and Hans Rullgård. Amoebas of maximal area. Internat. Math. Res. Notices, (9):441–451, 2001. | DOI | Zbl

[20] H. N. V. Temperley and Michael E. Fisher. Dimer problem in statistical mechanics—an exact result. Philos. Mag. (8), 6:1061–1063, 1961. | DOI | MR | Zbl

[21] Glenn Tesler. Matchings in graphs on non-orientable surfaces. J. Combin. Theory Ser. B, 78(2):198–231, 2000. | DOI | MR | Zbl

[22] L. G. Valiant. The complexity of computing the permanent. Theoret. Comput. Sci., 8(2):189–201, 1979. | DOI | MR | Zbl

Cited by Sources: